Ben Foster(Stanford) will give a talk at 09/10/2024

Given a harmonic polynomial on some bounded domain, how large can any of its level sets be? What about the critical set where its gradient vanishes? Considering simple examples shows that the size must depend on the degree of the polynomial, and for general solutions to elliptic equations, there is a quantity called the frequency function which measures the local growth of solutions and works as a substitute for the notion of degree. In two dimensions, connections with complex analysis can be exploited, and I’ll introduce a rich family of weighted estimates called Carleman inequalities. In higher dimensions, I’ll survey a few results relying on induction on scales and discuss some open questions of interest in this field. I’ll focus on the harmonic case throughout the talk for simplicity, with a brief discussion of the additional difficulties that arise when considering more general elliptic operators with coefficients that are less regular.